Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result of Borel-Serre, one shows that $\mathbb{Z}$ has a bounded resolution with finite free terms as $\mathbb{Z}[\Gamma]$-module (see for example section 5.1). This implies that $\Gamma$ has finite cohomological dimension. Moreover, this also shows that taking group cohomology commutes with filtered colimits.
The questions are the following:
Does the pro-completion $\hat{\Gamma}$ of $\Gamma$ have either finite cohomological dimension or the compatibility with filtered colimits?
Assume that $\Gamma$ is a congruence subgroup. Does the ''congruence'' pro-completion (i.e. the completion with respect to congruence subgroups of finite index) have finite cohomological dimension or compatibility with filtered colimits?
In particular, I would like to know if filtered colimits commute with continuous cohomology of the Iwahori subgroup of $\operatorname{GL}_2(\mathbb{Z}_p)$.
